# Binomial Theorem Questions

We provide binomial theorem practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on binomial theorem skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom. #### List of binomial theorem Questions

Question NoQuestionsClass
1The number of integral terms in the expansion of ( left(5^{1 / 2}+7^{1 / 8}right)^{1024} ) is11
2Coefficient of ( boldsymbol{x}^{k},(mathbf{0} leq boldsymbol{k} leq boldsymbol{n}) ) in
expansion of ( boldsymbol{P}=mathbf{1}+(mathbf{1}+boldsymbol{x})+(mathbf{1}+ )
( boldsymbol{x})^{2} ldots ldots+(1+boldsymbol{x})^{n} )
A. ( ^{n} C_{k} )
B. ( ^{n+1} C_{n-k-1} )
c. ( ^{n} C_{n-k} )
D. ( ^{n+1} C_{k+1} )
11
3Find the coefficient of ( x^{-7} ) in the
expansion of ( left(boldsymbol{a} boldsymbol{x}-frac{mathbf{1}}{boldsymbol{b} boldsymbol{x}^{2}}right)^{11} )
( begin{array}{ll}text { A. } & ^{11} C_{5}end{array} )
В. ( ^{10} C_{4} )
c. ( ^{11} C_{4} )
D. ( ^{10} C_{5} )
11
4Number of irrational terms in the
expansion of ( left(5^{frac{1}{6}}+2^{frac{1}{8}}right)^{100} )
A . 96
B. 97
c. 98
D. 99
11
5The sum of the coefficient of first 3
terms in the expansion ( left(x-frac{3}{x^{2}}right)^{m} ) in
( 559 . ) Find the term of the expansion containing ( boldsymbol{x}^{mathbf{3}} )
11
6If ( A ) and ( B ) are coefficients of ( x^{n} ) in the
expansions of ( (1+x)^{2 n} ) and ( (1+x)^{2 n-1} )
respectively, then ( frac{A}{B} ) is equal to
A .4
B . 2
( c .9 )
D. 6
11
7The coefficient of three consecutive
terms in the expansion of ( (1+a)^{n} ) are
in ratio 1: 7: 21 , then find the value of
( boldsymbol{n} )
11
8If ( C_{r} ) denotes the binomial coefficient
( ^{n} C_{r} ) then ( (-1) C_{0}^{2}+2 C_{1}^{2}+5 C_{2}^{2}+ )
( ldots(3 n-1) C_{n}^{2}= )
A ( cdot(3 n-2)^{2 n} C_{n} )
( ^{text {В }} cdotleft(frac{3 n-2}{2}right)^{2 n} C_{n} )
c. ( (5+3 n)^{2 n} C_{n} )
D ( cdotleft(frac{3 n-5}{2}right)^{2 n} C_{n+1} )
11
9Find ( a ) if the ( 17^{t h} ) and ( 18^{t h} ) term of the
expanse on ( (2+a)^{50} ) are equal.
11
10V1V9 VO101010101 011
The coefficient ofx7 in the expansion of (1-x-x2 + x) is

(a) -132
(b) -144
() 132
(d) 144
11
11The sum of all the coefficient of those
terms in the expansion of ( (a+b+c+d)^{8} ) which contains ( b ) but
not ( boldsymbol{c} ) is
( mathbf{A} cdot 6305 )
B ( cdot 4^{8}-3^{8} )
C. Number of ways of forming 8 digit numbers using digits 1,2,3 each number as atleast one 3
D. Number of ways of forming 4 digit numbers using digits 1,2,3 each number as atleast one 3
11
12( frac{C_{0}}{1}+frac{C_{1}}{2}+frac{C_{2}}{3}+ldots ldots+frac{C_{10}}{11}= )
A ( cdot frac{2^{11}}{11} )
B. ( frac{2^{11}-1}{11} )
c. ( frac{3^{11}}{11} )
D. ( frac{3^{11}-1}{11} )
11
13(a)
4
(b)
120
25.
If the coefficents of x3 and x4 in the expansion of
powers of x are both zero, then
(a, b) is equal to:
[JEE M 2014]
(a) (14,272)) (10,272) (c) (16,251) (a) (14,251)
11
14Number of rational term is the
expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{729} )
( A cdot 81 )
B. 82
c. 730
D. None of these
11
15If ( (1+x)^{n}=sum_{i=0}^{n} C_{i} x^{i}, ) then the sum of
the products of ( C_{i} ) ‘s taken two at a time is represented by ( sum_{0 leq i leq j leq n} C_{i} C_{j} )
A ( cdot 2^{n}-frac{(2 n) !}{2(n !)^{2}} )
B. ( 2^{n}+frac{(2 n) !}{2(n !)^{2}} )
c. ( frac{1}{2}left(2^{2 n}+frac{(2 n) !}{(n !)^{2}}right) )
D. ( frac{2^{2 n}}{2(n !)^{2}} )
11
16The middle term in the expansion of ( left(frac{a}{x}+b xright)^{12} )
A ( cdot 924 a^{6} b^{6} )
B. ( 924 a^{6} b^{5} )
( mathbf{c} cdot 924 a^{5} b^{5} )
D. ( 924 a^{5} b^{6} )
11
17The number of rational terms in the
expansion of ( left(mathbf{9}^{1 / 4}+mathbf{8}^{1 / 6}right)^{1000} ) is:
A . 500
в. 400
( c .501 )
D. none of the above
11
18A positive integer which is just greater ( operatorname{than}(1+0.0001)^{10000} ) is
( A cdot 3 )
B. 4
( c .5 )
D. 6
11
19et n be positive integer. If the coefficients of 2nd, 3rd, and
1th terms in the expansion of (1 + x)” are in A.P., then the
value of n is …………
(1994 – 2 Marks)
11
20Find ( A_{2}^{n}, ) if the fifth term of the
expansion of ( left(sqrt{x}+frac{1}{x}right)^{n} ) does not
depend on ( boldsymbol{x} )
11
21The coefficient of the middle term in the
expansion of ( (1+x)^{2 n} ) is
This question has multiple correct options
A ( cdot 2^{n} C_{n} )
в. ( frac{1.3 .5 ldots ldots(2 n-1)}{n !} 2^{text {। }} )
c. ( 2.6 ldots(4 n-2) )
D ( cdot 2.4 ldots ldots . .2 n )
11
22The Coefficient of ( x^{n} ) in the expansion of
( (1+x)(1-x)^{n} ) is
A. ( (n-1) )
B ( cdot(-1)^{n-1} n )
C ( cdot(-1)^{n-1}(n-1)^{2} )
D・ ( (-1)^{n}(1-n) )
11
23Find ( (boldsymbol{a}+boldsymbol{b})^{4}-(boldsymbol{a}-boldsymbol{b})^{4} ). Hence,
evaluate ( (sqrt{mathbf{3}}+sqrt{mathbf{2}})^{4}-(sqrt{mathbf{3}}-sqrt{mathbf{2}})^{4} )
11
24If ( left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}+mathbf{1}right)^{boldsymbol{6}}=boldsymbol{a}_{boldsymbol{0}}+left(boldsymbol{a}_{boldsymbol{1}} boldsymbol{x}+frac{boldsymbol{b}_{boldsymbol{1}}}{boldsymbol{x}}right)+ )
( left(a_{2} x^{2}+frac{b_{2}}{x^{2}}right)+ldots+left(a_{6} x^{6}+frac{b_{6}}{x^{6}}right) ) the
find the value of ( a_{0} )
11
25The coefficient of ( x^{3} ) in ( left(sqrt{x^{5}}+frac{3}{sqrt{x^{3}}}right)^{6} )
is
( mathbf{A} cdot mathbf{0} )
в. 120
( c cdot 420 )
D. 540
11
26Which term in the expansion of ( (1+x)^{p} cdotleft(1+frac{1}{x}right)^{q} ) is independent of ( x )
where ( p, q ) are positive integers? What is the value of that term?
11
27Find the coefficient of ( x^{-17} ) in the
expansion of ( left(x^{4}-frac{1}{x^{3}}right)^{15} )
A. 1200
B. -1331
c. -1365
D. -2016
11
28Expand the following expression in
ascending powers of ( x ) as far as ( x^{3} ) ( frac{1+2 x}{1-x-x^{2}} )
11
29( (1+x)^{21}+(1+x)^{22}+ldots+(1+x)^{30} )
coefficient of ( boldsymbol{x}^{5} )
11
30The term independent of ( x ) in the expansion of ( left(sqrt{frac{x}{3}}+frac{3}{2 x^{2}}right)^{10} ) will be
( ^{A} cdot frac{3}{2} )
в.
( c cdot frac{5}{2} )
D. None of these
11
3113. If the coefficient of x’
equals the
(bx)
in
coefficient of x-
the relation
(a) a-b=1
, then a and b satisfy

(b) a+b=1
(d) ab=1
11
3226. The sum of coefficients of integral power of x in the binomial
expansion (1-27x) is:
[JEE M 2015
(b) (250 +1)
11
33If the constant term in the binomial
expansion of ( left(x^{2}-frac{1}{x}right)^{n}, n quad epsilon quad N ) is 15
then the value of ( n ) is equal to
11
34In the binomial expansion of ( (a- )
( b)^{n}, n geq 5, ) the sum of
5 th and 6 th terms is zero then a/b equal
to
A ( cdot frac{5}{n-4} )
B. ( frac{6}{n-5} )
c. ( frac{n-5}{6} )
D. ( frac{n-4}{5} )
11
35Expand
(i) ( (sqrt{3}+sqrt{2})^{4} )
11
36Binomial expansion of ( left(boldsymbol{x}^{k}+frac{mathbf{1}}{mathbf{2}^{mathbf{2 k}}}right)^{mathbf{3 n}} )
where ( n ) is a positive integer, always contains a term which is independent of
( mathbf{A} cdot x^{2} )
B. ( x )
( mathbf{c} cdot x^{3} )
D. none of the above
11
37Expand the following binomial ( left(1-3 a^{2}right)^{6} )11
38The coefficient of ( x^{4} ) in the expansion of
( left(frac{x}{2}-frac{3}{x^{2}}right)^{10} ) is equal to:
11
39Write general term of this:-
( 2 xleft(3+2 x^{2}right)^{20} )
11
40Solve : ( left(3 x-frac{1}{2 y}right)left(3 x+frac{1}{2 y}right) )11
41The number of terms whose values
depends on ( x ) in the expansion of ( left(x^{2}-2+frac{1}{x^{2}}right)^{n} ) is
( mathbf{A} cdot 2 n+1 )
B. ( 2 n )
( c )
D. none of these
11
4210. The coefficient of x^ in expansion of (1 + x) (1 – x)” is
(a) (-1)”-In
(b) (-1)” (1-n) [2004
(c) (−1)n-1(n-1) (d) (n-1)
11
43The middle term of ( left(boldsymbol{x}-frac{1}{boldsymbol{x}}right)^{2 n+1} ) is
( mathbf{A} cdot^{2 n+1} C_{n} cdot x )
B. ( 2 n+1 C_{n} )
C ( cdot(-1)^{n} cdot 2^{2+1} C_{n} )
D ( cdot(-1)^{n} cdot^{2 n+1} C_{n} cdot x )
11
44Coefficient of ( x^{5} ) in ( left(1+x^{2}right)^{5}(1+x)^{4} ) is
A . 60
B. 80
( c cdot 90 )
D. 100
11
45Show that the middle term in the
expansion of ( (1+x)^{2 n} ) is
( frac{1.3 .5 ldots .(2 n-1)}{mid underline{n}} 2^{n} x^{n} )
11
46The term independent of ( x ) in ( (1+ )
( x)^{n}left(1+frac{1}{x}right)^{n} ) is
A ( cdot C_{0}^{2}+2 cdot C_{1}^{2}+3 cdot C_{2}^{2}+ldots ldots ldots+(n+1) cdot C_{n}^{2} )
B. ( left(C_{0}+C_{1}+C_{2}+ldots ldots . .+C_{n}right)^{2} )
c. ( C_{0}^{2}+C_{1}^{1}+C_{2}^{2}+ldots ldots . .+C_{n}^{2} )
D. None
11
47Find ( x, ) if it is known that the second
term of the expansion of ( left(x+x^{log x}right)^{5} ) is
equal to 1000000
11
48Find the ( 13^{t h} ) term in the expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18} )11
49Find the value of a given ( mathbf{3}+frac{mathbf{1}}{mathbf{4}}(mathbf{3}+boldsymbol{p})+frac{mathbf{1}}{mathbf{4}^{2}}(mathbf{3}+mathbf{2} boldsymbol{p})+frac{mathbf{1}}{mathbf{4}^{3}}(mathbf{3}+ )
( mathbf{3} boldsymbol{p})+ldots=? )
11
50The term independent of ( x ) in the expansion of
( left(x^{2}+frac{1}{x}right)^{9} ) is
( A )
B. –
c. 48
D. 84
11
51Expand ( left(x^{2}+frac{3}{x}right)^{4} )11
52The ratio of fifth term from the
beginning to the fifth term from the end in the expansion of ( left(sqrt{2}+frac{1}{sqrt{3}}right)^{n} ) is
( sqrt{6}: 1 . ) If ( n=frac{20}{lambda}, ) find the value of ( lambda )
11
53Find the coefficient of ( x^{5} ) in the product
( (1+2 x)^{6}(1-x)^{7} ) using binomial
theorem.
11
5430. If the fourth term in the Binomial expansion of ( = + xlog8x
(x>0) is 20×87, then a value of x is:
(JEEM 2019-9 April (M)
(a) 8 (6) 8 (c) 8 (d) 82
11
55The coefficient of ( x^{160} ) in the expansion
of ( left(x^{8}+right. )
1) ( ^{60}left(x^{12}+3 x^{4}+frac{3}{x^{4}}+frac{1}{x^{12}}right)^{-10} )
A. ( ^{30} C_{6} )
B. ( ^{30} C_{5} )
c. divisible by 189
D. divisible by 203
11
56In the expansion of ( (x+sqrt{x^{2}-1})^{6}+ )
( (x-sqrt{x^{2}-1})^{6}, ) the number of terms is
A. 7
B. 14
( c cdot 6 )
D.
11
57Which term in the expansion of ( left(frac{x}{3}-frac{2}{x^{2}}right)^{10} ) contains ( x^{4} ? )
( A )
B. 3
( c cdot 4 )
D.
11
58Expand using Binomial Theorem ( left(1+frac{x}{2}-frac{2}{x}right)^{4}, x neq 0 )11
59If rth term in the expansion of
( left(x^{2}+frac{1}{x}right)^{12} ) is independent of ( x, ) then
( boldsymbol{r}= )
( mathbf{A} cdot mathbf{9} )
B. 8
c. 10
D. none of these
11
60Find the square of the following binomials by using the identity ( (-z+6) )11
61The value of
( frac{18^{3}+7^{3}+3.18 .7 .25}{3^{6}+6.243 .2+15.81 .4+20.27 .8+15.9 .16+6.3 .32+64} )
( A cdot 4 )
( B .3 )
( c cdot 2 )
D.
11
62The coefficient of ( x^{4} ) in the expansion of
( {sqrt{1+x^{2}}-x}^{-1} ) in ascending powers
of ( x, ) when ( |x|<1 ) is
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{2} )
( c cdot-frac{1}{2} )
D. ( -frac{1}{8} )
11
63The sum of the last eight coefficients in
the expansion of ( (1+x)^{15}, ) is ( ? )
A ( cdot 2^{16} )
B . ( 2^{15} )
( c cdot 2^{14} )
D. none of these
11
64if the coefficient of the middle term in
the expansion of ( (1+x)^{2 n+2} ) and ( p ) and
the coefficients of middle terms in the
expansion of ( (1+x)^{2 n+1} ) are ( q ) and ( r )
then
A ( . p+q=r )
в. ( p+r=q )
c. ( p=q+r )
D. ( p+q+r=0 )
11
65If the first three terms in the expansion
of ( (1+a x)^{n} ) are ( 1,8 x, 24 x^{2} )
respectively, then ( a= )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D.
11
66The coeff. of ( 8^{t h} ) term in the expansion of
( (1+x)^{10} ) is
A. 120
B. 7
c. ( ^{10} C_{8} )
D. 210
11
67If the coefficient of 4 consecutive terms
in the expansion of ( (1+x)^{n} ) are
( a_{1}, a_{2}, a_{3}, a_{4} ) respectively, then show
that:
( frac{boldsymbol{a}_{mathbf{1}}}{boldsymbol{a}_{mathbf{1}}+boldsymbol{a}_{mathbf{2}}}+frac{boldsymbol{a}_{mathbf{3}}}{boldsymbol{a}_{mathbf{3}}+boldsymbol{a}_{boldsymbol{4}}}=frac{mathbf{2} boldsymbol{a}_{boldsymbol{3}}}{boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{boldsymbol{3}}} )
11
686.
mial expansion of (a – b)”, n25, the sum of the 5th
and 6th terms is zero. Then alb equals
(20015)
In the hi
(a)(n-5) 16
(0)55-4)
(6) [m-415
(d) Gen 5)
11
69Find the coefficient of ( x^{50} ) in the
expression:
( (1+x)^{1000}+2 x(1+x)^{999}+ )
( mathbf{3} x^{2}(mathbf{1}+boldsymbol{x})^{mathbf{9} 9 mathbf{8}}+ldots .+mathbf{1 0 0} mathbf{1} boldsymbol{x}^{mathbf{1 0 0 0}} )
A ( .^{1000} mathrm{C}_{50} )
В. ( ^{1001} mathrm{C}_{50} )
( mathbf{c} cdot^{1002} C_{50} )
D. ( ^{1003} Omega_{50} )
11
70In the expansion of ( left(sqrt{4}+frac{1}{sqrt{6}}right)^{20} )
This question has multiple correct options
A. the number of rational terms ( =4 )
B. the number of irrational terms ( =19 )
C. the middle term is irrational
D. the number of irrational terms ( =17 )
11
71Expand ( (1-2 x)^{5} )11
72r and n are positive integers r> 1, n > 2 and coefficient of
(r+2)th term and 3rth term in the expansion of (1 + x)2n are
equal, then n equals

(2) 3r (6) 3r+1 (c) 2r (d) 2r+1
11
73If the fourth term in the expansion of ( left(sqrt{frac{1}{x^{log x+1}}}+x^{1 / 12}right)^{6} ) is equal to 200
and ( x>1, ) then ( x ) is equal to
A ( cdot 10^{sqrt{2}} )
2 ( sqrt{2} )
B. 10
( c cdot 10^{4} )
D. None of these
11
74Coefficient of ( boldsymbol{x}^{mathbf{5 0}} )
( (x>0), ) in ( (1+x)^{1000}+2 x(1+ )
( boldsymbol{x})^{999}+mathbf{3} boldsymbol{x}^{2}(mathbf{1}+boldsymbol{x})^{998}+ldots ) is
A. ( 1000 C_{50} )
B. ( ^{1000} C_{50} )
c. ( ^{1002} 250 )
D. ( ^{1000} C_{49} )
11
75The coefficient of the middle term in the
binomial expansion in power of ( x ) of
( (1+alpha x)^{4} ) and of ( (1-alpha x)^{6} ) is the same
if ( alpha ) equals-
A ( cdot-frac{5}{3} )
в. ( frac{10}{3} )
( c cdot frac{-3}{10} )
D.
11
76In the expansion of the expression ( (x+a)^{15}, ) if the eleventh term in the
geometric mean of the eighth and
twelfth terms, which term in the
expression is the greatest?
A. ( T_{6} )
в. ( T_{7} )
c. ( T_{8} )
D. ( T_{9} )
11
77The ( 3 r d, 4 t h ) and 5 th terms in the
expansion of ( (1+x)^{n} ) are 60,160 and
240 respectively, then ( x= )
( A cdot 2 )
B. 4
( c .5 )
D. 6
11
78Assertion
( (sqrt{2}-1)^{n} ) can be expressed as ( sqrt{N} ) ( sqrt{N-1} ) for ( forall N>1 ) and ( n in N )
Reason
( (sqrt{2}-1)^{n} ) can be written in the form ( boldsymbol{alpha}+boldsymbol{beta} sqrt{boldsymbol{2}} forall, boldsymbol{alpha}, boldsymbol{beta} ) are integers & n is a
positive integer.
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
c. Assertion is true but Reason is false
D. Assertion is false but Reason is true.
11
79The number of terms which are free
from radical signs in the expansion of ( left(y^{frac{1}{5}}+x^{frac{1}{10}}right)^{55} ) are
A. 5
B. 6
( c cdot 7 )
D. none of these
11
80Find the cube of the following binomial expressions:
( frac{3}{x}-frac{2}{x^{2}} )
11
81If ( |x|<1 ) then the coefficient of ( x^{n} ) in
expansion of ( left(1+x+x^{2}+x^{3} dotsright)^{2} ) is
( A )
B . ( n-1 )
( mathbf{c} cdot n+2 )
( mathbf{D} cdot n+1 )
11
82If the constant term of the binomial zpansion ( left(2 x-frac{1}{x}right)^{n} ) is -160 , then ( n ) is equal to
A .4
B. 6
( c cdot 8 )
D. 10
11
83The term in dependent of ( x ) in ( left(1+x+2 x^{3}right)left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} )
A ( cdot frac{25}{54} )
в. ( frac{17}{54} )
( c cdot frac{1}{6} )
D. ( -frac{17}{51} )
11
84Find the 7 th term from the end in the
expansion of ( left(2 x^{2}-frac{3}{2 x}right)^{8} )
11
85If ( n ) is an integer between 0 and 21 , then
the minimum value of ( n !(21-n) ! ) is
attained for ( n= )
A .
B. 10
c. 12
D. 20
11
86The sum of the co-efficients of all odd
degree terms in the expansion of ( (x+sqrt{x^{3}-1})^{5}+(x+sqrt{x^{3}-1})^{5} )
( (x>1) ) is:
A . 2
B. –
( c cdot 0 )
( D )
11
87Find the ( 13^{t h} ) terms in the expansion of
( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 )
A . 18564
B. 87328
c. 17374
D. 35546
11
88Write the general term in the expansion of ( left(x^{2}-y^{2}right)^{6} )11
89The term independent of ( x ) in ( left(frac{1}{2} x^{frac{1}{3}}+right. )
( left.boldsymbol{x}^{frac{-1}{5}}right)^{8} ) is
A. ( frac{35}{8} )
B. 7
c. ( frac{7}{2} )
D . 28
11
90Find the middle term(s) in the
expansion of :
( left(2 a x-frac{b}{x^{2}}right)^{12} )
A ( cdot frac{59136 a^{6} b^{6}}{x^{6}} )
В. ( frac{59163 a^{5} b^{5}}{x^{5}} )
c. ( frac{59631 a^{7} b^{7}}{x^{7}} )
D. None of these
11
91Which number is larger (1.1) ( ^{100000} ) or
( mathbf{1 0}, mathbf{0 0 0} ? )
11
92If ( a ) and ( b ) are distinct integers, prove
that ( a-b ) is a factor of ( a^{n}-b^{n} )
whenever ( n ) is a positive integer.
11
93Find ( 7^{t h} ) term of ( left(frac{4 x}{5}-frac{5}{2 x}right)^{9} )
A. ( frac{10050}{x^{3}} )
в. ( frac{10500}{x^{3}} )
c. ( frac{1050}{x^{3}} )
D. ( frac{1000}{x^{3}} )
11
94Given positive integers ( i>1, n>2 ) and that the coefficients of ( (3 r)^{t h} ) and ( (r+ )
2) ( ^{t h} ) terms in the bionomial expansion
of ( (1+x)^{2 n} ) are equal, then
A ( . n=2 r )
B. n=3r
c. ( n=2 r+1 )
D. None of these.
11
95Given positive integers r>1, n >2 and that the coefficient of
(3r)th and (r + 2)th terms in the binomial expansion of
(1+ x)2n are equal . Then
(1983 – 1 Mark)
(a) n=2r
(c) n=2r+1
(c) n=3r
(d) none of these
11
96The number of dissimilar terms in the
expansion of ( left(1-3 x+3 x^{2}-x^{3}right)^{20} ) is
A . 21
B. 32
c. 41
D. 61
11
97The sum of the coefficients in the first
three terms of the expansion of ( left(x^{2}-frac{2}{x}right)^{m} ) is equal to ( 97 . ) Find the term of the expansion containing ( boldsymbol{x}^{4} )
11
98In the expansion of ( (a+b)^{n}, ) the ratio of
the binomial coefficients of ( 2^{n d} ) and ( 3^{r d} )
terms is equal to the ratio of the binomial coefficients of ( 5^{t h} ) and ( 4^{t h} )
terms, then ( n= )
A . 4
B. 5
( c .6 )
D.
11
99The term independent of ( x ) in ( (2 x- ) ( left.frac{1}{2 x^{2}}right)^{12} ) is
( mathbf{A} cdot-^{12} C_{3} cdot 2^{6} )
B. ( -^{12} C_{5} .2^{2} )
c. ( 12 C_{6} )
D. ( ^{12} C_{4} .2^{4} )
11
100If ( n ) is a positive integer and ( (5 sqrt{5}+ )
( mathbf{1 1})^{2 n+1}=I+f ) where I is an integer
and ( 0<f<1 ) then
This question has multiple correct options
A. I is an even integer
B. ( (I+f)^{2} ) is divisible by ( 2^{2 n+1} )
c. lis divisible by 22
D. None of these
11
101The coefficient of the term independent of ( x ) in the expansion of
( left(frac{x+1}{x^{frac{2}{3}}-x^{frac{1}{3}}+1}-frac{x-1}{x-x^{frac{1}{2}}}right)^{10} )
( A cdot 70 )
в. 112
c. 105
D. 210
11
102Find the middle term in the expansion of ( left(frac{2 x^{2}}{3}-frac{3}{2 x}right)^{12} )11
103If in the expansion of ( (1-x)^{2 n-1}, ) the
coefficient of ( x^{r} ) denoted by ( a_{r}, ) then :
( mathbf{A} cdot a_{r-1}+a_{2 n-r}=0 )
В ( cdot a_{r-1}-a_{2 n-r}=0 )
c. ( a_{r-1}+2 a_{a n-r}=0 )
D. None of these
11
104If ( (1+x)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+dots+ )
( a_{10} x^{10}, ) then the value of
( left(a_{0}-a_{2}+a_{4}-a_{6}+a_{8}-a_{10}right)^{2}+ )
( left(a_{1}-a_{3}+a_{5}-a_{7}+a_{9}right)^{2} ) is
( mathbf{A} cdot 2^{10} )
B . 2
( c cdot 2^{20} )
D. None of these
11
105The ( 4 t h ) term from the end in the
expansion of ( left(frac{x^{3}}{2}-frac{2}{x^{2}}right)^{7} ) is
A . ( 35 x )
B. ( 70 x^{2} )
( c cdot 35 x^{2} )
D. ( 70 x )
11
106The number of rational terms in the expansion of ( left(3 frac{1}{4}+7 frac{1}{6}right)^{144} )
A . 33
B. 23
( c cdot 12 )
D. 13
11
107If the 20 th and 21 st terms in the
expansion of ( (1+x)^{40} ) are equal, then
the value of ( x ) is
A ( cdot frac{20}{21} )
в. ( frac{21}{20} )
c. 25
D. ( frac{1}{25} )
11
108Find the term which has the exponent of ( x ) as 8 in the expansion of
( left(x^{frac{5}{2}}-frac{3}{x^{3} sqrt{x}}right)^{10} )
( mathbf{A} cdot T_{2} )
В. ( T_{3} )
c. ( T_{4} )
D. Does not exist
11
109For ( boldsymbol{r}=mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, dots, mathbf{1 0}, ) let ( boldsymbol{A}_{r}, boldsymbol{B}_{r}, boldsymbol{C}_{boldsymbol{r}} )
denote respectively the coefficient of ( x^{r} )
in the expansions of ( (1+x)^{10},(1+x)^{20} )
and ( (1+x)^{30} . ) Then ( sum_{r=1}^{10} A_{r}left(B_{10} B_{r}-right. )
( left.C_{10} A_{r}right) ) is equal to
( mathbf{A} cdot B_{10}-C_{10} )
B . ( A_{10}left(B_{10}^{2}right)-C_{10} A_{10} )
c.
D. ( C_{10}-B_{10} )
11
110Number of irrational terms in the binomial expansion of ( left(3^{1 / 5}+7^{1 / 3}right)^{100} )
is
A . 94
B. 88
c. 93
D. 95
11
111Show that the coefficient of ( a^{m} ) and ( a^{n} )
in the expansion of ( (1+a)^{m+n} ) are
equal.
11
112Given that the ( 4^{t h} ) term in the expansion of ( left(2+frac{3 x}{8}right)^{10} ) has the maximum numerical value, then ( x ) can lie in the
interval(s)
This question has multiple correct options
( mathbf{A} cdotleft(2, frac{64}{21}right) )
B ( cdotleft(-frac{60}{23},-2right) )
( mathbf{C} cdotleft(-frac{64}{21},-2right) )
D ( cdotleft(2,-frac{60}{23}right) )
11
113In any binomial expansion, the number
of terms are
( A cdot geq 5 )
B. ( geq 2 )
( c cdot geq 3 )
( D cdot geq 4 )
11
114In the expansion of ( left(boldsymbol{a} sqrt{boldsymbol{a}}+frac{mathbf{1}}{boldsymbol{a}^{4}}right)^{boldsymbol{n}} ), the
coefficient in the second term exceeds
by 44 the coefficient in the first term. Find ( n )
A . 20
B . 25
( c .35 )
D. 45
11
115If the sum of the coefficients of ( x^{2} ) and
coefficients of ( x ) in the expansion of
( (1+x)^{m}(1-x)^{n} ) is equal to ( -m, ) then
the value of ( 3(n-m) ) is
(Note ( : boldsymbol{m}, boldsymbol{n} text { are distinct }) )
11
116Using Binomial theorem, evaluate ( (mathbf{9 9})^{5} )11
117The product of two middle terms in the
expansion of ( left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} ) is
( ^{mathrm{A}} cdotleft(^{9} C_{4}right)^{2} cdot frac{x^{9}}{512} )
в. ( -9_{C_{4}} .^{9} C_{5}, frac{x^{8}}{512} )
c. ( _{-9}^{text {g }} q_{4} .^{9} C_{5} ). ( frac{x^{9}}{512} )
D・ ( _{9} C_{4} .^{9} C_{5}, frac{x^{9}}{256} )
11
118Solve ( (1+i)^{4}+(1-i)^{4}= )11
119Find the expansion of ( (boldsymbol{a}-boldsymbol{2} boldsymbol{x})^{boldsymbol{7}} )11
120In the expansion of ( left(x^{3}-frac{1}{x^{2}}right)^{n}, n in N )
if the sum of the coefficient of ( x^{5} ) and
( x^{10} ) is ( 0, ) then ( n ) is
A . 25
B. 20
c. 15
D. None of these
11
121Multiply the binomials.
( (y-8) ) and ( (3 y-4) )
11
122Find the middle term in the expansion of ( (5 x-7 y)^{7} )11
12315. If the expansion in powers of x of the function
is ao +ajx+azx? +azx?… then a, is
(1 – ax)(1-bx)
6″ -ah
b-a

(b) a” – 6”
b-a
bn+1-an+1
b-a
an+l – 11+1
b-a
11
124Let ( n ) be a positive integer such that
( left(1+x+x^{2}right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+ )
( ldots+a_{2 n} x^{2 n}, ) then ( a_{r}= )
В ( cdot a_{2 n} ; 0 leq r leq 2 n )
D. None of these
11
125The expansion ( left[boldsymbol{x}+left(boldsymbol{x}^{mathbf{3}}-mathbf{1}right)^{mathbf{1} / mathbf{2}}right]^{mathbf{5}}+[boldsymbol{x}- )
( left.left(x^{3}-1right)^{1 / 2}right]^{5} ) is a polynomial of degree
A. 5
B. 6
( c cdot 7 )
( D )
11
126If the sum of the coefficients in the
expansion of ( (a+b)^{n} ) is ( 4096, ) then the
greatest coefficient in the expansion is
A ( cdot 924 )
в. 792
( c .1594 )
D. None of these
11
127In the expansion of ( (1+x)^{n}, ) the ( 5^{t h} )
term is 4 times the ( 4^{t h} ) term and the ( 4^{t h} )
term is 6 times the ( 3^{r d} ) term. than ( n= )
( A cdot 9 )
B. 10
( c cdot 11 )
D. 12
11
128Find the 7 th term from the end in the
expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 )
11
129The number of terms with integral coefficients in the expansion of
( left(7^{1 / 3}+5^{1 / 2} cdot xright)^{600} ) is
A. 100
B. 50
( c .101 )
D. none of these
11
130The coeffcient of ( x^{10} ) in the expansion of
( (1+x)^{2}left(1+x^{2}right)^{3}left(1+x^{3}right)^{4} ) is equal to
A . 52
B. 44
c. 50
D. 56
11
1313. If (itaa)? 1 +880 +2438 t… thena…. andon…11
132Find the middle terms of the equation of
( left(x^{4}-frac{1}{x^{3}}right)^{11} )
( mathbf{A} cdot-462 x^{9}, 462 x^{2} )
B . ( -462 x^{8}, 462 x^{4} )
c. ( 462 x^{7},-462 x^{3} )
D. None of these
11
133In the expansion of ( (sqrt{2}+sqrt{5})^{20} ) the
number of rational terms will be:
( A cdot 3 )
B. 10
( c cdot 4 )
D.
11
134Find the coefficient of ( x^{5} ) and ( x^{-15} ) in the
expansion of ( left(3 x^{2}-frac{1}{3 x^{3}}right)^{10} ? )
11
135The total number of terms in the
expansion of ( (x+a)^{100}+(x-a)^{100} )
after simplification is
A .202
B. 51
( c .50 )
D. 49
11
136Find the middle term(s) of ( left(frac{x^{3 / 2} y}{2}+right. )
( left.frac{2}{x y^{3 / 2}}right)^{13} )
11
137Expand ( left(x^{2}+2 aright)^{5} ) by binomial
theorem.
11
138Expand the binomial ( left(frac{2 x}{3}+frac{3 y}{2}right)^{20} u p ) to four terms.11
1397.
The number of integral terms in the expansion of
(13+ 5)256 is

(a) 3 (6) 32 (6) 33 (d) 34
11
140For ( mathbf{r}=mathbf{0}, mathbf{1}, dots, 10, ) let ( mathbf{A}_{mathbf{r}}, mathbf{B}_{mathbf{r}} ) and ( mathbf{C}_{mathbf{r}} )
denote, respectively, the coefficient of ( x^{r} ) in the expansions of ( (1+x)^{10},(1+ )
( mathbf{x})^{20} ) and ( (mathbf{1}+mathbf{x})^{30} ) Then ( sum_{r=1}^{10} boldsymbol{A}_{r}left(boldsymbol{B}_{10} boldsymbol{B}_{r}-right. )
( left.C_{10} A_{r}right) ) is equal to
A. ( mathrm{B}_{10}-mathrm{C}_{10} )
B . ( A_{10}left(B_{10}^{2}-C_{10} A_{10}right) )
c. 0
D. ( C_{10}-B_{10} )
11
141Find the middle terms in the expansion
of ( (5 x-7 y)^{7} )
11
142(0
)
10
10
Let S = { j (j – 1)!°C,, S, = \$ 710c; and S3 = 2,2 10C
j=1
j=1
statement-1:S = 55 x 29.
Statement-2: S, =90 x 2

nt-2: S = 90 x 28 and S. = 10 x 28.
Statement – 1 is true. Statement -2 is true ; Statement-2
not a correct explanation for Statement-1.
Statement -1 is true, Statement -2 is false.
Statement – 1 is false, Statement -2 is true.
statement – lis true, Statement 2 is true; Statement -2
1
1
.
c…datamant –
11
143If ‘p’ and ‘q’ are the coefficients of ( x^{a} ) and
( x^{b} ) respectively in ( (1+x)^{a+b}, ) then
A. ( 2 p=q )
В. ( p+q=0 )
c. ( p=q )
D. ( p=2 q )
11
14411.
For r=0, 1, …, 10, let A, B and C, denote, respectively,
the coefficient of x’ in the expansions of (1 + x)”, (2010)
10
(1 + x)20 and (1 + x)30. Then ZA(B10B.-C104,) is equal to
(a) B10-C10
(b) A10(B216C10410
(d) C10-B10
11
145Find the middle term(s) in the
expansion of :
( left(3 x-frac{x^{3}}{6}right)^{9} )
A ( cdot frac{189}{8} x^{15},-frac{21}{16} x^{17} )
В ( cdot frac{189}{8} x^{17},-frac{21}{16} x^{19} )
C. ( frac{189}{7} x^{15},-frac{23}{13} x^{19} )
D. None of thes
11
146The coefficient of ( boldsymbol{x}^{r}[mathbf{0} leq boldsymbol{r} leq boldsymbol{n}-mathbf{1}] ) in
the expression of ( (x+2)^{n-1}+(x+ )
2) ( ^{n-2} cdot(x+1)+(x+2)^{n-3} cdot(x+1)^{2}+ )
( ldots+(x+1)^{n-1} ) is
A ( cdot^{n} C_{r}left(2^{r}-1right) )
B. ( ^{n} C_{r}left(2^{n-r}-1right) )
c. ( ^{n} C_{r}left(2^{r}+1right) )
D. ( ^{n} C_{r}left(2^{n-r}+1right) )
11
147la
J
in the expansion of (1 + x)” (1 – x)”, the coefficients of x
od r2 are 3 and – 6 respectively, then mis (1999 – 2 Marks)
a) 6 (6) 9
(c) 12 (d) 24
11
14816.
b-a
For natural numbers m, nif (1-y)” (1 + y)”
=1+ay +a,y2 + ……. and a, = a, = 10, then (m, n) is
(a) (20,45)
(b) (35,20)

(c) (45,35)
(d) (35,45)
ceth
11
149Find the middle term of ( left(frac{a}{x}+frac{x}{a}right)^{10} )11
150The coefficient of ( x^{30} ) in the expansion of
( left(1+2 x+3 x^{2}+dots .21 x^{20}right)^{2} ) is
A . 2706
в. 2450
( c .1481 )
D. 256
11
151If it is known that the third term of the
binomial expansion ( left(x+x^{log _{10} x}right)^{3} ) is ( 10^{6} )
then ( x ) is equal to
A . 10
B. ( 10^{frac{5}{2}} )
c. 100
D. 5
11
152The coefficient of ( x ) in ( left(x^{2}+frac{c}{x}right)^{5} ) is
A . 20
B. 10
( c cdot 10 c^{3} )
D. 20 ( c^{3} )
11
153State the whether given statement is true or false
Prove that the coefficient of xnxn
A. True
B. False
11
15428. The value of
(21C, -10C,)+(°C, – 10C,)+(!Cz – 1°C3)+(1C4 – 10C)
+…+(+1C70-10C10) is:
[JEE M 2017
(a) 220 -210
(b) 221 – 211
(c) 221 – 210
(d) 220 – 29
11111
11
155If in the expansion of ( left(frac{1}{x}+x tan xright)^{5}, ) the ratio of ( 4^{t h} ) term to the ( 2^{n d} ) term is ( frac{2}{27} pi^{4} ) then the value of ( x ) can be
A ( cdot frac{-pi}{6} )
в. ( frac{-pi}{3} )
c.
D. ( frac{pi}{12} )
11
156If the coefficients of ( a^{m} ) and ( a^{n} ) in the
expansion of ( (1+a)^{m+n} ) are ( alpha ) and ( beta ) then which one of the following is correct?
A ( cdot alpha=2 beta )
в. ( alpha=beta )
c. ( 2 alpha=beta )
11
157The value of ( ^{n} C_{0}+3 times^{n} C_{1}+9 times^{n} )
( boldsymbol{C}_{2}+ldots+boldsymbol{3}^{n} times^{n} boldsymbol{C}_{n} )
A ( cdot 2^{n} )
B. ( 3^{n} )
( c cdot 4^{n} )
D. ( 5^{n} )
11
158(1982 – 2 Marks)
The sum of the coefficients of the plynomial (1+x -3×2 2163
is ………
(1982-2 Marlo
IF(1 Iarn -119.242
11
159The ( 3 r d, 4 t h, ) and 5 th terms in the
expansion ( (x+a)^{n} ) are respectively
( 84,280, ) and ( 560, ) find the values of ( x, a )
and ( n )
A. ( x=1, a=2, n=6 )
B. ( x=1, a=6, n=7 )
c. ( x=3, a=2, n=7 )
D. ( x=1, a=2, n=7 )
11
160Find the term independent of ( ^{prime} x^{prime} ) in the expansion of the expression, ( (1+x+ )
( left.2 x^{3}right)left(frac{3}{2} x^{2}-frac{1}{3 x}right)^{9} )
11
161ff ( f(x)=x^{4}+10 x^{3}+39 x^{2}+76 x+65 )
find the value of ( f(x-4) )
11
162Find the coeffcient of:
( x^{-7} ) in the expansion of ( left(a x-frac{1}{b x^{2}}right)^{8} )
ii) ( x^{6} ) in the expansion ( left(a-b x^{2}right)^{10} )
11
163If the fourth term in the expansion of
( (p x+1 / x)^{n} ) is ( 5 / 2 ) then the value of ( p )
is
( A )
B. 1/2
( c cdot 6 )
D.
11
164The term independent of ( x ) in the expansion of ( left(1+x+2 x^{3}right)left(frac{3 x^{2}}{2}-right. )
( left.frac{1}{3 x}right)^{9} ) is
A ( cdot frac{13}{54} )
в. ( frac{15}{54} )
c. ( frac{17}{54} )
D. ( frac{19}{54} )
11
165The coefficient of ( x^{5} ) in the expansion of
( left(1+x^{2}right)^{5}(1+x)^{4} ) is ( ? )
( mathbf{A} cdot 61 )
B. 59
c. zero
D. 60
11
166The middle term of expansion of ( left(frac{10}{x}+frac{x}{10}right)^{10} )
A ( cdot^{7} C_{5} )
в. ( ^{8} C_{5} )
( mathrm{c} cdot^{9} mathrm{C}_{5} )
D. ( ^{10} C_{5} )
11
167Find the term of expansion of ( left(x+frac{1}{x}right)^{n} ) which does not contain ( x )11
168Let ( n ) be a positive integer. If the
coefficients of ( 2^{n d}, 3^{r d} ) and ( 4^{t h} ) terms in
the expansion of ( (1+x)^{n} ) are in A.P.
then the value of ( n ) is:
( mathbf{A} cdot mathbf{8} )
B. 27
c. 12
D.
11
169Write the middle terms in the
expansion of ( left(frac{3 x}{7}-2 yright)^{10} )
11
170( (sqrt{3}+sqrt{2})^{4}-(sqrt{3}-sqrt{2})^{4}= )
A ( .20 sqrt{6} )
в. ( 30 sqrt{6} )
c. ( 5 sqrt{10} )
D. ( 40 sqrt{6} )
E ( .10 sqrt{6} )
11
171The greatest value of the term independent for ( x ) in the expansion of
( left(x sin p+x^{-1} cos pright)^{10}, p in R, ) is
( A cdot 2^{5} )
в. ( frac{10 !}{(5 !)^{2}} )
c. ( frac{1}{2^{5}} cdot frac{10 !}{(5 !)^{2}} )
D. None of the above
11
172distinct primes, then show that In nk ln2 (1984-2 Marks)
Find the sum of the series :
1 31 7
C – +- +- +-
Ar ….. up to m terms]
27 22r
Č (–19 “C,13+ 2 + 2 + y pu.. up to m terms]
237 x 15
24r ….. Un
11
173|
(4)
O2
(U)
– 1
23. If n is a positive integer , then (13+1)?” -(13 – 1)?” is:

(a) an irrational number
(6) an odd positive integer
an even positive integer
(d) a rational number other than positive integers
(C)
an even
11
174Number of terms free from radical sign in the expansion of ( left(1+3^{1 / 3}+7^{1 / 2}right)^{10} )
is
( A cdot 4 )
B. 5
( c cdot 6 )
D.
11
175Find the value(s) of k such that the term
independent of ( x ) in ( left(3 x^{2}+frac{k}{2 x}right)^{6} ) is 135
( A cdot pm 2 )
B. ±1
( c .pm 3 )
D. ±4
11
176Let ( [x] ) denote the greatest integer part of a real number x. If
( boldsymbol{M}=sum_{n=1}^{40}left[frac{boldsymbol{n}^{2}}{mathbf{2}}right] )
then m equals
A . 5700
B. 5720
( c .5740 )
D. 11060
11
177Sum of coefficients in the expeansion of
( (a+b+c)^{8} ) is
A. 2154
в. 6561
c. 729
D. 1944
11
178The sum of the coefficients of first three
terms in the expansion of ( left(x-frac{3}{x^{2}}right)^{m}, x neq 0, m ) being a natura
number, is ( 559 . ) Find the term of the
expansion containing ( boldsymbol{x}^{mathbf{3}} )
11
179Find the negative of middle term in the expansion of
( left(frac{2 x}{3}-frac{3}{2 x}right)^{6} )
11
180The first 3 terms in the expansion of ( (1+a x)^{n}(n neq 0) ) are ( 1,6 x ) and ( 16 x^{2} )
Then the value of ( a ) and ( n ) are
respectively
A . 2 and 9
B. 3 and 2
c. ( 2 / 3 ) and 9
D. ( 3 / 2 ) and 6
11
181If ( C_{0} . C_{1}, C_{2}, dots, C_{n} ) are the coefficients
of the expansion of ( (1+x)^{n}, ) then the value of ( sum_{0}^{n} frac{C_{k}}{k+1} ) is
A .
в. ( frac{2^{n}-1}{n} )
c. ( frac{2^{n+1}-1}{n+1} )
D. None of these
11
182( operatorname{Let}left(1+x^{2}right)^{2}(1+x)^{n}=A_{0}+A_{1} x+ )
( A_{2} x^{2}+ldots . ) If ( A_{0}, A_{1}, A_{2} ) are in A.P, then
the value of ( n )
( A cdot 2 )
B. 3
( c cdot 5 )
D. 7
11
183The middle term in the expansion of
( left(frac{x}{y}+frac{y}{x}right)^{8} ) is
( A cdot^{8} C_{5} )
в. ( ^{8} mathrm{C}_{6} )
( mathrm{c} cdot^{8} mathrm{C}_{4} )
D. ( ^{8} mathrm{C}_{2} )
11
1842
4
27. If the number of terms in the expansion of 1-+-
X+0, is 28, then the sum of the coefficients of all the term
in this expansion, is :
JJEEM 2016
(a) 243 (b) 729 (c) 64 (d) 2187
11
185Find the middle term in the expansion
of :
( left(frac{x}{a}-frac{a}{x}right)^{10} )
11
186Find the greater number in ( 300 ! ) and ( sqrt{300^{300}} )11
187Find the number of terms in the
expansion of ( left(1-2 x+x^{2}right)^{7} )
11
188The term independent of ( x ) in the expansion of ( left(x^{2}-frac{3 sqrt{3}}{x^{3}}right)^{10} )11
189Using binomial theorem find the value of ( (102)^{3} )11
190Find the term independent of ( x ) in ( (x+ ) ( left.frac{1}{x}right)^{4} )11
19114. If x is so small that rand higher powers of x may be
(1 + x)2 –
1+
neglected, then
may be approximated as
(1-x)2
(a) 1-3×2
(b) 3x + 2×2

8
12
to
8
11
1926.
If x is positive, the first negative term in the expansion of
(1+x) 27/5 is

(a) 6th term (b) 7th term (c) 5th term (d) 8th term.
11
193The value of the expansion ( (sqrt{3}+1)^{5}+ )
( (sqrt{3}-1)^{5} )
( mathbf{A} cdot 88 )
B . 40
c. ( 88 sqrt{3} )
D. ( 40 sqrt{3} )
11
194Find the second term of the binomial
expansion of ( left(sqrt{boldsymbol{a}}+frac{boldsymbol{a}}{sqrt{boldsymbol{a}^{-1}}}right)^{m}, ) if ( boldsymbol{C}_{3}^{boldsymbol{m}} )
( C_{2}^{m}=4: 1 )
11
195The third term from the end in the
expansion of ( left(frac{4 x}{3 y}-frac{3 y}{2 x}right)^{9} ) is
A ( cdot ) s ( _{C_{7}} frac{3^{5}}{2} frac{y^{5}}{x^{5}} )
В. ( -_{-9} sigma_{7} frac{3^{5}}{2^{3}} frac{y^{5}}{x^{5}} )
c. ( _{9} C_{7} frac{3^{5}}{2^{3}} frac{y^{5}}{x^{3}} )
D. none of these
11
196Using Binomial Theorem, evaluate ( (101)^{4} )11
197( left(begin{array}{l}n \ 0end{array}right)+2left(begin{array}{l}n \ 1end{array}right)+2^{2}left(begin{array}{l}n \ 2end{array}right)+ldots .+2^{n}left(begin{array}{l}n \ nend{array}right) )
is equal to
A ( cdot 2^{n} )
B.
( c cdot 3^{n} )
D. None of these
11
198If the coefficients of rth, (r+1)th, and (r+2)th terms in the
the binomial expansion of (1+y)” are in A.P., then mand
r satisfy the equation

(a) m? – m(4r-1)+4r2 – 2 = 0
(b) m2 – m (4r+1)+4 r2 +2=0
(c) m2 – m (4r+1)+ 4 r2 – 2=0
(d) m? –m (4r-1)+4 2 +2 = 0
11
199If in the expansion of ( (1+x)^{43}, ) the
coefficient of ( (2 r+1) t h ) term is equal
to coefficient of ( (r+2)^{t h} ) term. Find ( r ) ??
11
200( operatorname{Let}left(frac{2 x^{2}+x+2}{x}right)^{n}=sum_{r=m}^{r=t} a_{r} x^{r} )
( fleft(a_{p}=a_{q} text { then } p+q=dotsright. )
11
201The coefficient of ( x^{5} ) in the expansion of
( left(x^{2}-x-2right)^{5} ) is
A. 351
в. -82
c. -86
D. -81
11
202If ( boldsymbol{X}=left{4^{n}-3 n-1: n in Nright} ) and
( boldsymbol{Y}={mathbf{9}(boldsymbol{n}-mathbf{1}): boldsymbol{n} in boldsymbol{N}}, ) where ( mathrm{N} ) is
the set of natural numbers, then ( boldsymbol{X} cup boldsymbol{Y} )
is equal to:
A. ( Y )
B. ( N )
c. ( Y-x )
D. ( x )
11
203Write down and simplify:
The 25 th term of ( (5 x+8 y)^{30} )
11
204The coefficient of ( x ) in the expansion of ( (1-a x)^{-1}(1-b x)^{-1}(1-c x)^{-1} ) is?
A ( . a+b+c )
B. ( a-b-c )
c. ( -a+b+c )
D. ( a-b+c )
11
205Find the middle term in the expansion of ( left(frac{x}{a}-frac{a}{x}right)^{21} )
A ( cdot 20_{110} frac{x}{a},^{21} C_{10} frac{a}{x} )
В. ( _{20} C_{9} frac{x}{a},^{2} 16_{10} frac{a}{x} )
( ^{mathrm{C}} cdot_{21} C_{10} frac{x}{a^{2}},-^{21} 1_{10} frac{a}{x} )
D. ( _{21} C_{9} frac{x}{a},^{21} C_{10} frac{a}{x} )
11
206f ( x+y=1, ) then ( sum_{r=0}^{n} r^{n} C_{r} x^{r} . y^{n-r}= )
( mathbf{A} cdot mathbf{1} )
B.
c. ( n x )
D. ( n y )
11
207The total number of terms in the
expansion of ( (x+y)^{50}+(x-y)^{50} ) is
A . 51
B . 26
( c .102 )
D. 25
11
208Show that the middle term in the
expansion of ( (1+x)^{2 n} ) is ( frac{1.3 .5 ldots . . .(2 n-1)}{n !} )
( 2^{n} x^{n} ; ) where ( n ) is a positive integer.
11
209The third term from the end in the
expansion of ( left(frac{3 x}{5}-frac{5}{2 x}right)^{8} ) is
A ( cdot frac{35451}{15 x^{4}} )
в. ( frac{45455}{16 x^{4}} )
c. ( frac{39372}{15 x^{4}} )
D. ( frac{39375}{16 x^{4}} )
11
210Expand the following binomial ( left(1-frac{1}{x}right)^{10} )11
211( operatorname{If}left(1+2 x+x^{2}right)^{n}=sum_{r=0}^{2 n} a_{r} x^{r}, ) then ( a_{r}= )
A ( cdotleft(^{n} C_{r}right)^{2} )
В. ( ^{n} C_{r} cdot^{n} C_{r+1} )
( c cdot^{2 n} C_{r} )
D. ( ^{2 n} C_{r+1} )
11
212Number of irrational terms in the expansion of ( (sqrt{2}+sqrt{3})^{15} ) is equal to
A . 16
B. 7
c. 12
D. 15
11
213The expansion of ( left[boldsymbol{x}+left(boldsymbol{x}^{mathbf{3}}-mathbf{1}right)^{mathbf{1} / 2}right]^{mathbf{5}}+ )
( left[x-left(x^{3}-1right)^{1 / 2}right]^{5} ) is a polynomial of
degree
( mathbf{A} cdot mathbf{8} )
B. 7
( c cdot 6 )
D. 5
11
214In the expansion of ( left(3^{-x / 4}+3^{5 x / 4}right)^{n} )
the sum of binomial coefficient is 64
and term with the greatest bionomial coefficient term exceeds the third term
by ( (n-1) ) the value of ( x ) must be
( A cdot 0 )
B.
( c cdot 2 )
D. 3
11
215Assertion
No three consecutive binomial
coefficient can be in G.P. & H.P.
Reason
Three consecutive binomial coefficients
are in A.P.
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
c. Assertion is true but Reason is false
D. Assertion is false but Reason is true
11
216Suppose that ( n ) is a natural number and
( boldsymbol{I}, boldsymbol{F} ) are respectively the integral part
and fractional part of ( (7+4 sqrt{3})^{n} . ) Then
show that
i) ( I ) is an odd integer
ii) ( (boldsymbol{I}+boldsymbol{F})(mathbf{1}-boldsymbol{F})=mathbf{1} )
11
217If the 7 th and 8 th term of the binominal
expansion ( (2 a-3 b)^{n} ) are equal, then ( frac{2 a+3 b}{2 a-3 b} ) is to
A ( cdot frac{n-13}{n+1} )
в. ( frac{n+1}{13-n} )
c. ( frac{6-n}{13-n} )
D. ( frac{n-1}{13-n} )
11
218In the binomial expansion of ( (1+y)^{n} )
where ( n ) is a natural number, the
coefficients of the ( 5^{t h}, 6^{t h} ) and ( 7^{t h} ) terms
are in A.P, find ( n )
This question has multiple correct options
( mathbf{A} cdot n=7 )
B . ( n=14 )
c. ( n=8 )
D. ( n=16 )
11
219The value of
( left(10^{11}-10^{9}-2 times 11 times 10^{8}-3 times 11^{2} timesright. )
is equal to
( mathbf{A} cdot mathbf{0} )
B. ( 11^{10} )
( c cdot 11^{11} )
D. ( 10^{11} )
11
220In the expansion of ( left(x-frac{1}{x}right)^{6} ), the
constant term is
A . -20
B . 20
c. 30
D. -30
11
221The coefficient of the term independent of ( x ) in the expansion of ( left(frac{x+1}{x^{frac{2}{3}}-x^{frac{1}{3}}+1}-frac{x-1}{x-x^{frac{1}{2}}}right)^{10} )
A . 210
в. 105
c. 70
D. 112
11
222If the fourth term in the expansion of
( left(sqrt{frac{1}{x log x+1}}+x frac{1}{12}right)^{6} ) is equal to
200 and ( x>1, ) then ( x ) is
A . 10
B. ( 10^{-4} )
( c cdot 1 )
D. -4
11
223The co-efficient of ( x^{53} ) in the expression ( sum_{m=0}^{100} 100 c_{m}(x-3)^{100-m} 2^{m} ) is
B. ( 98_{c_{53}} )
( mathbf{c} .^{65} sigma_{53} )
D. ( 100 c_{65} )
11
224The coefficient of ( t^{24} ) in the expansion of
( left(1+t^{2}right)^{12}left(1+t^{12}right)left(1+t^{24}right) ) is
A. ( ^{12} C_{6}+2 )
B. ( ^{12} C_{5} )
( mathbf{c} cdot^{12} C_{6} )
D. ( ^{12} C_{7} )
11
225Find the ( (p+2) ) th term from the end in ( left(x-frac{1}{x}right)^{2 n+1} )11
226The value of ( x ) in the expression
( left(x+x^{log _{10} x}right)^{5}, ) if the third term in the
expansion is ( 10,00,000, ) is This question has multiple correct options
A ( cdot 10^{-1} )
B . ( 10^{text {। }} )
( mathbf{c} cdot 10^{-5 / 2} )
D. ( 10^{5 / 2} )
11
227For positive integers ( n_{1}, n_{2} ) the value of
the expression ( (1+i)^{n_{1}}+left(1+i^{3}right)^{n_{1}}+ )
( left(1+i^{5}right)^{n_{2}}+left(1+i^{7}right)^{n_{2}}, i=sqrt{-1} ) is a
real number if and only if
( mathbf{A} cdot n_{1}=n_{2}+1 )
В . ( n_{1}=n_{2}-1 )
c. ( n_{1}=n_{2} )
D ( . n_{1}>0, n_{2}>0 )
11
228If ( boldsymbol{X}=left{mathbf{8}^{n}-mathbf{7} boldsymbol{n}-mathbf{1}, boldsymbol{n} in boldsymbol{N}right} ) and ( boldsymbol{Y}= )
( mathbf{4} 9(n-1), n in N, ) then ( (operatorname{given} n>1) )
A. ( X subset Y )
в. ( Y subset X )
c. ( X=Y )
D. ( X nsubseteq Y )
11
229The positive integerjust greater than
( (1+0.0001)^{10000} ) is
( A cdot 4 )
B. 5
( c cdot 2 )
D.
11
230The sum of the coefficients in the first,
second, and third terms of the expansion of ( left(x^{2}+frac{1}{x}right)^{m} ) is equal to 46 Find the term of the expansion which does not contain ( x )
11
231If ( (1+a x)^{n}=1+8 x+24 x^{2}+dots ) then
( boldsymbol{a} times boldsymbol{n} ) is:
( mathbf{A} cdot mathbf{8} )
B. 12
c. 16
D . 24
11
232Evaluate ( (sqrt{3}+sqrt{2})^{6}-(sqrt{3}-sqrt{2})^{6} )11
23319. Let n be a positive integer and
(1994 – 5
(1 + x + x2)n = a+a, x+ …………+ a), x2
Show that a 2-a,2+ a,2 ………….+ a,,2=an
11
234Find the expansion ( left(3 x^{2}-2 a x+3 a^{2}right)^{3} )
using binomial theorem.
11
235The value of the term independent of
( x ) in the expansion of ( left(x^{2}-frac{1}{x}right)^{27} ) is :
( mathbf{A} cdot mathbf{9} )
B. 18
c. 48
D. 84
11
236The number of integral terms in ( (sqrt{3}+sqrt{2})^{64} ) is
( mathbf{A} cdot mathbf{8} )
B. 7
( c .9 )
D. 6
11
237If for ( 1 leq m leq n, f(m, n)=C_{0}-C_{1}+ )
( C_{2}-ldots .(-1)^{m-1} C_{m-1}, ) find ( f(9,5) )
11
238( 5^{t h} ) term from the end in the expansion
of ( left(frac{x^{2}}{2}-frac{2}{x^{2}}right)^{12} ) is
B . ( 7920 x^{4} )
c. ( 7920 x^{-4} )
D. ( -7920 x^{4} )
11
239The term independent of ( x ) in the expansion of ( left(x-frac{1}{x}right)^{4}left(x+frac{1}{x}right)^{3} ) is
A . -3
B.
( c .3 )
D.
11
240Find the fourteenth term of ( (3-a)^{15} )11
241If the coefficient of three consecutive
terms in the expansion of ( (1+x)^{n} ) be
( 165,330, ) and ( 462 . ) Find ( n )
11
242( (1-sqrt{2})^{6}= )
A ( .98-70 sqrt{2} )
В. ( 99-70 sqrt{2} )
D. ( 98+70 sqrt{2} )
11
2439.
The coefficient of the middle term in the binomial expansion
in powers of x of (1+ ax)4 and of (1 – ax) is the same if a
equals

(a)
5 (6)
(b)
(c) To (d) –
11
244The coefficient of ( x^{9} ) in the expansion of ( left(x^{3}+frac{1}{2^{t}}right)^{11}, ) where ( t=log _{sqrt{2}}left(x^{frac{3}{2}}right) )
A . -5
в. 330
( c .520 )
D. ( 5+log _{sqrt{2}}(3) )
11
245Find the term independent of ( x ) in the expansion of ( left(sqrt{frac{boldsymbol{x}}{mathbf{3}}}+frac{mathbf{3}}{mathbf{2} boldsymbol{x}^{2}}right)^{10} )
( mathbf{A} cdot T_{3} )
в. ( T_{4} )
c. ( T_{5} )
D. No term will be independent of ( x )
11
246In the expansion of ( left(x^{3}-frac{1}{x^{2}}right)^{2}, ) where n is a positive integer, the sum of the
coefficients of ( boldsymbol{x}^{boldsymbol{6}} ) is ( mathbf{1} )
11
247The 2 nd, 3 rd and 4 th terms in the
expansion of ( (x+y)^{n} ) are 240,720,1080 respectively; find ( x, y, n )
11
248Solve ( :left(frac{2 n}{2 n-1}right)^{p}=left(frac{1}{1-left(frac{p}{2 n}right)}right)^{p} )11
249( f^{n-1} C_{r}=left(k^{2}-3right)^{n} C_{r+1}, ) then ( k )
B. ( [2, infty) )
c. ( [-sqrt{3}, sqrt{3}] )
D. ( (sqrt{3}, 2] )
11
250In the expansion of ( left(x^{2}-frac{1}{4}right)^{n}, ) the coefficient of third term is ( 31, ) then the
value of ( n ) is-
A . 30
B. 31
c. 29
D. 32
11
251ff ( left(x^{2}+frac{1}{x}right)^{n} ) has exactly one middle term which is equal to ( alpha . x^{3} ) then the
value of ( (boldsymbol{alpha}+boldsymbol{n}) ) is- ( quad(boldsymbol{n} in boldsymbol{N}) )
A . 18
B . 21
c. 24
D. 26
11
2523.
The positive integer just greater than (1 +0.0001)10000 is

(2) 4 (6) 5 (c) 2 (d) 3
11
253Expand
( left(2 x^{2}+3right)^{4} )
11
254If ( (5+2 sqrt{6})^{n}=m+f ), where ( n ) and ( m )
are positive integers and ( 0 leq f<1 )
then
( frac{1}{1-f}-f ) is equal to
A ( cdot frac{1}{m} )
в. ( m )
c. ( _{m+frac{1}{m}} )
D. ( _{m-} frac{1}{m} )
11
255If the third term in the expansion of ( left[x+x^{log _{10} x}right]^{5} ) is ( 10^{6}, ) then ( x ) can be
This question has multiple correct options
A ( cdot 10^{-1 / 3} )
B. 10
c. ( 10^{-5 / 2} )
D. ( 10^{2} )
11
256The middle term in the expansion of
( left(frac{2 x}{3}-frac{3}{2 x^{2}}right)^{2 n} ) is
A ( cdot 2^{n} mathrm{C}_{n} )
B cdot ( (-1)^{n}left[(2 n !) /(n !)^{2}right] cdot x^{-n} )
( mathrm{c} cdot_{2 n} mathrm{C}_{n} cdot frac{1}{x^{n}} )
D. none of these
11
257Expansion of ( (boldsymbol{y}+boldsymbol{x})^{n} ) is11
258The number of rational terms in the
expansion of ( left(x^{frac{1}{5}}+y^{frac{1}{10}}right)^{45} ) is
A. 5
B. 6
( c cdot 4 )
D.
11
259If ( T_{r} ) denotes the ( r^{t h} ) term in the
expansion of ( left(x+frac{1}{x}right)^{23}, ) then
( mathbf{A} cdot T_{12}=T_{13} )
В . ( x^{2} . T_{13}=T_{12} )
c. ( x^{2} . T_{12}=T_{13} )
D. ( T_{12}+T_{13}=25 )
11
260e sum of the co-efficients of all odd degree terms in the
expansion of
(x+Vx3 -1)3 +(x-Vx3-1)”,(x>1) is : [JEEM 2018]
(a) o (6) 1 do (c) 2
(d) – 1
11
261The middle term in the expansion of ( left(x+frac{1}{x}right)^{10} )
A ( cdot ) io ( _{1} frac{1}{x} )
в. ( ^{10} C_{5} )
c. ( ^{10} C_{6} )
D. ( ^{10} C_{7} x )
11
262Find the middle term in the expansion of ( left(frac{2 x}{3}+frac{3}{2 x}right)^{10} )
A . 210
в. 630
( c .252 )
D. 756
11
263UL. TUU lay uuu
23.
Let n be any positive integer. Prove that
(1999 – 10 Marks)
2n-k
mlk
n-k)
2n-2m
(2n-4k+1) 9n-2k =
on-Ak
(2n-2k +1) (2n – 2m)
In-m
kao
for each non-be gatuve integer msn. | Here
11
264Find the sixth term in the expansion of
( left(2 x^{2}-frac{3}{7 x^{3}}right)^{11} )
A ( cdot-^{11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{3} )
В ( cdot quad^{11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{-3} )
c. ( _{-11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{-3} )
D. None of these
11
265Find coefficient of ( a^{3} b^{4} c^{5} ) in the
expansion of ( (b c+c a+a b)^{6} )
11
266In the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{6561} )
prove that three will be only 730 term which are free from radicals
11
267In the expansion of ( (1+x)^{n}, ) the
binomial coefficients of 3 consecutive
terms are respectively 220,495 and 792
then ( n= )
A .4
B. 8
( c cdot 12 )
D. 16
11
268If the constant term of the binomial expansion ( left(2 x-frac{1}{x}right)^{n} ) is ( -160, ) then ( n ) is equal to –
A .4
B. 6
c. 8
D. 10
11
269Expand to 4 terms the following expressions:
( (1+x)^{frac{2}{5}} )
11
270f ( log 1001=3.000434 ), find the number of digits in ( 1001^{101} )11
271Expansion of ( (3 x+2)^{3} ) is ( 27 x^{3}+8+ )
( 18 x(3 x+2) )
A. True
B. False
11
272The tenth term in the expansion of
( left(2 x^{2}+frac{1}{x}right)^{12} )
A ( cdot frac{1760}{x^{3}} )
в. ( -frac{1760}{x^{3}} )
c. ( frac{1760}{x^{2}} )
D. none of the above
11
2733
17.
145, 35)
In the binomial expansion of (a – b)”, n > 5, the sum of

and 6th terms is zero, then a/b equals
n-5 a n-4 o 5 (d)
(a)”
n-5
20071
11
274Write general terms of this
( 2 xleft[3+2 x^{2}right]^{20} )
11
275The coefficient of ( x^{11} ) in the expansion of
( left(1-2 x+3 x^{2}right)(1+x)^{11} ) is
A . 164
в. 144
c. 116
D. none of these
11
276The coeficient of x’in (
The coefficient of x4 in
is
(1983 – 1 Mark)
“(19831 Mar
(b) 504
504
(a)
405
256
259
450
263
(d) none of these
11
2771200T
18.
The sum of the series
20 Co – 20G + 2002 – 2003 +… -.+ 206, is
(a) o (6) 20. c) _2000 (a) 220610
11
278Expand ( left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}right)^{boldsymbol{6}} )11
279Find the middle term(s) in the
expansion of ( left(1+3 x+3 x^{2}+x^{3}right)^{2 n} )
11
280If ( left(1+x^{2}right)^{2}(1+x)^{n}=C_{0}+C_{1} x+ )
( C_{2} x^{2}+cdots, ) and if ( C_{0}, C_{1}, C_{2} ) are in A.P.
find ( n )
This question has multiple correct options
A .2
B. 3
( c cdot 4 )
D. 5
11
281The middle term in the expansion of
( left(x+frac{1}{x}right)^{10} )
A ( cdot ) io ( C_{4} cdot frac{1}{x} )
в. ( ^{10} C_{5} )
c. ( _{10} mathrm{C}_{5} . frac{1}{x} )
D. ( ^{10} C_{6} . x )
11
282The greatest value of the term independent of ( x, ) as ( alpha ) varies over ( R ) in
the expansion of ( left(x cos alpha+frac{sin alpha}{x}right) ) is
B. ( ^{20} C_{19} )
c. ( 20 C_{6} )
D. ( ^{20} C_{6}left(frac{1}{2}right)^{10} )
11
283If the sum of the coefficients in the expansion of (a + b)” is
4096, then the greatest coeficient in the expansion is

(a) 1594 (b) 792 (c) 924 (d) 2924
anni 10000.
11
284Find the middle term in the expansion
of ( left(1-2 x+x^{2}right)^{n} )
A ( cdot frac{(2 n) !}{(n !)^{2}}(-1)^{n} x^{n} )
B. ( frac{(2 n) !}{(n !)}(-1)^{n} x^{2} n )
c. ( frac{(2 n) !}{(n !)^{2}} x^{n} )
D. None of these
11
285Find the coefficients of ( x^{2} ) and ( x^{3} ) in the
expansion of ( (2-x)^{6} )
11
286Write the general term in the expansion
of ( left(x^{2}-yright)^{6} )
11
287The ratio of coefficient of ( x^{3} ) and ( x^{4} ) in
expansion ( (1+x)^{12} ) is:
( A cdot frac{4}{9} )
B . ( frac{1}{3} )
( c cdot frac{2}{3} )
D.
11
288και
η 5. Σε
αυτά της – Σ
ε
εθει με τον ερχολ το
ΥΞΟ
2n -1
(2004)
(c)
n – 1
11
289Which of the following binomial expressions has a least term
independent of ( x ? )
( ^{mathrm{A}} cdotleft(sqrt{x}-frac{3}{x^{2}}right)^{1} )
B. ( left(x+frac{1}{x}right)^{6} )
c. ( (1+x)^{32} )
( ^{mathrm{D}}left(frac{3}{2} x^{2}-frac{1}{3 x}right)^{9} )
11
290Find the coefficient of ( x^{17} ) in ( (x+y)^{20} ? )11
291If the middle term in the expansion of ( left(x^{2}+frac{1}{x}right)^{n} ) is ( 924 x^{6}, ) then ( n= )
A . 10
B. 12
c. 14
D. none of these
11
292The term independent of ( x ) in the expansion of ( left(boldsymbol{x}^{2}-frac{mathbf{1}}{boldsymbol{x}}right)^{boldsymbol{6}} ) is
A . -12
B. 15
( c cdot 24 )
D. -15
11
293The coefficient of ( t^{50} ) in
( left(1+t^{2}right)^{25}left(1+t^{25}right)left(1+t^{40}right)left(1+t^{45}right)(1 )
is
A ( cdot 1+sqrt{5} )
B. ( 1+^{25} C_{5}+^{25} C_{7} )
( mathbf{C} cdot 1+^{25} C_{7} )
D. None of these
11
294What is the unit digit in the product ( left(3^{65} times 6^{59} times 7^{71}right) ? )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. 6
11
295The middle term in the expansion of ( left(1-3 x+3 x^{2}-x^{3}right)^{2 n} ) is
( mathbf{A} cdot 6 n_{C_{3 n}}(-x)^{3 n} )
B . ( 6 n_{C_{2 text { n }}}(-x)^{2 n+1} )
( mathbf{c} cdot 4 n_{C_{S n}}(-x)^{3 n} )
D. ( 6 n_{C_{3 n}}(-x)^{3 n-1} )
11
296The middle term in the expansion of
( (1+x)^{2 n} ) is
A. ( frac{1.3 .5 ldots(2 n-1)}{n} x^{n} )
B. ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n-1} x^{n} )
c. ( frac{1.3 .5 ldots(2 n-1)}{n !} x^{n} )
D. ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n} x^{n} )
11
297Find the middle term in the expansion
of
( left(frac{2}{3} x-frac{3}{2 x}right)^{20} )
A ( .^{20} C_{10} x^{10} y^{10} )
B. ( ^{20} C_{11} x^{11} y^{11} )
C. ( ^{20} C_{9} x^{11} y^{10} )
D. None of these
11
298f ( ^{n} C_{4},^{n} C_{5},^{n} C_{6} ) of ( (1+x)^{n} ) are in A.P.
then ( n= )
A . 12
B. 1
( c cdot 7 )
( D .8 )
11
299Expand the expression ( (2 x-3)^{6} )11
300Coefficient of ( boldsymbol{x}^{boldsymbol{9}} ) in ( rightarrow(mathbf{1}+boldsymbol{x})(mathbf{( 1}+boldsymbol{t}) )
( left.left.boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{3}right) ldots ldotsleft(boldsymbol{1}+boldsymbol{x}^{mathbf{1 0 0}}right)right) ? )
11
30114. F E 4, (x-2)* = 6, (x – 3)” and ax = 1 for all
p=0
(1992 – 6 Marks)
k > n, then show that b, = 2NFC
11
30219.
Statement-1:
“C, = (n + 2)2n-1

r=0
Statement-2: E(r+1) “Cyx” = (1 + x)” + nx(1+x)”-.
r=0
(a) Statement-1 is false, Statement-2 is true
(b) Statement-1 is true, Statement-2 is true; Statement -2 15
a correct explanation for Statement-1
Statement -1 is true, Statement-2 is true; Statement -2
is not a correct explanation for Statement-1
(d) Statement -1 is true, Statement-2 is false
11
303Find the term of the expansion of ( (sqrt{x^{-2}}+x)^{7} ) containing ( x ) in the
second power
( mathbf{A} cdot T_{4} )
в. ( T_{5} )
c. ( T_{6} )
( mathbf{D} cdot T_{7} )
11
304The number of non-zero terms in the
expansion of ( (sqrt{7}+1)^{75}-(sqrt{7}-1)^{75} )
is
A . 36
B. 37
c. 38
D. 39
11
305( 9^{t h} ) term in the expansion of
( left(frac{x}{a}-frac{3 a}{x^{2}}right)^{12} )
A ( cdot^{12} C_{9} cdot 3^{9} x^{-12} a^{6} )
В. ( ^{12} C_{6} cdot 3^{8} x^{-16} a^{6} )
C ( .^{12} C_{4} cdot 3^{8} x^{-12} a^{4} )
D. none of the above
11
30624.
The term independent of x in expansion of
J
JEEM 2013
x+1_ -_*-1 is
(r2/3 – X1/3+1 x – x12
(a) 4
(b) 120
(C) 210
(d) 310
11
307The middle term in the expansion of
( left(frac{10}{x}+frac{x}{10}right)^{10} )
A . ( ^{10} C_{5} )
в. ( ^{10} C_{6} )
c. ( _{10} mathrm{C}_{5} frac{1}{x^{10}} )
D. ( ^{10} C_{5} x^{10} )
E . ( ^{10} C_{5} 10^{10} )
11
308Find the sum of the coefficients of the
terms of the expansion ( left(1+x+2 x^{2}right)^{6} )
11
309Show that the middle term in the
expansion of ( (1+x)^{2 n} ) is
( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n} x^{n}, ) where ( n ) is a
positive integer.
11
310The value of ( (sqrt{5}+1)^{5}-(sqrt{5}-1)^{5} ) is:
A . 252
в. 352
c. 452
D. 552
11
311Prove that the coefficient of ( x^{n} ) in the
expression of ( (1+x)^{2 n} ) is twice the
coefficient of ( x^{n} ) in the expression of
( (1+x)^{2 n-1} )
11
312The co-efficient of ( x^{5} ) in the expansion of
( (1+x)^{21}+(1+x)^{22}+dots dots+ )
( (1+x)^{30} ) is:
A . ( ^{51} C_{5} )
в. ( ^{9} C_{5} )
c. ( ^{31} C_{6}-^{21} C_{6} )
D. ( ^{30} C_{5}+^{20} C_{5} )
11
313Find the value ( (sqrt{3}+1)^{4}+(sqrt{3}-1)^{4}=? )11
314For ( boldsymbol{r}=mathbf{0}, mathbf{1}, mathbf{2},, dots mathbf{1 0} ) let ( boldsymbol{A}_{r}, boldsymbol{B}_{r} ) and ( boldsymbol{C}_{boldsymbol{r}} )
denote respectively the coefficient of ( boldsymbol{x}^{r} ) in the expansions of ( (1+x)^{10},(1+x)^{20} ) and ( (1+x)^{30} . ) Then
( sum_{r=1}^{10} A_{r}left(B_{10} B_{r}-C_{10} A_{r}right) ) is equal to
A. ( B_{10}-C_{10} )
B . ( A_{10}left(B_{10}^{2}-C_{10} A_{10}right) )
c.
D. ( C_{10}-B_{10} )
11
315If the second term of the expansion ( left[boldsymbol{a}^{1 / 13}+frac{boldsymbol{a}}{sqrt{boldsymbol{a}^{-1}}}right]^{n} quad boldsymbol{i s} 14 boldsymbol{a}^{5 / 2}, ) then the
value of ( frac{n_{mathbf{S}}}{n_{mathbf{C}_{2}}} ) is
A .4
B. 3
c. 12
D. 6
11
316In the expression of ( left(2^{x}+frac{1}{4^{x}}right)^{n} ) ratio of
2nd and third terms is given by ( t_{3} / t_{2}= )
7 and the sum of the co-efficients of 2 nd
and ( 3 mathrm{rd} ) term is ( 36, ) then the value of ( x )
is
A ( frac{-1}{3} )
в. ( frac{-1}{2} )
( c cdot frac{1}{3} )
D.
11
317If ( omega neq 1 ) is a cube root of unity and
( (omega+x)^{n}=1+12 omega+69 omega+ldots . ) then
values of ( 4 n ) and ( omega ) respectively are
A . 36,1
в. 12,2
c. ( 24,1 / 2 )
D. ( 18,1 / 3 )
11
318In the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{6561} ) prove that there will be only 730 terms which are free from radicals.11
319If the coefficients of ( x ) and ( x^{2} ) in the
expansion of ( (1+x)^{m}(1-x)^{n} ) are 3
and -6 respectively. Find the values of
( m ) and ( n )
11
320The coefficient of ( t^{4} ) in the expansion of
( left(1+t^{2}right)^{3} )
( A )
B. 3
( c cdot-3 )
D. –
11
321The highest term in the expansion of ( (2 sqrt{5}+sqrt{7})^{6} ) is
A. ( 800 sqrt{35} )
55
в. ( 700 sqrt{35} )
c. ( 320 sqrt{5} )
D. ( 100 sqrt{7} )
11
322If ( boldsymbol{f}(boldsymbol{x})=(mathbf{1}+boldsymbol{x})^{15}=boldsymbol{C}_{0}+boldsymbol{C}_{1} boldsymbol{x}+ )
( C_{2} x^{2}+ldots+C_{15} x^{15}, ) then ( f(2)= )
( mathbf{A} cdot 1^{15} )
B. ( 3^{15} )
( c cdot 2^{15} )
D. None of these
11
323Consider the expansion ( left(x^{2}+frac{1}{x}right)^{15} ) Consider the following statements:
1. There are 15 terms in the given expansion.
2. The coefficient of ( x^{12} ) is equal to that
of ( x^{3} )

Which of the statements is/are correct
( ? )
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2

11
324Find the ( 4^{t h} ) term in the expansion of
( (x-2 y)^{12} )
11
325If ( c_{0}, c_{1}, c_{2}, ldots ldots c_{n} ) are the coefficients in
the expansion of ( (1+x)^{n}, ) when ( n ) is a
positive integer, prove that
(1) ( c_{0}-c_{1}+c_{2}-c_{3}+dots dots+ )
( (-1)^{r} c_{r}=(-1)^{r} frac{mid n-1}{|underline{r}| n-r-1} )
(2) ( c_{0}-2 c_{1}+3 c_{2}-4 c_{3}+dots dots+ )
( (-1)^{n}(n+1) c_{n}=0 )
(3) ( c_{0}^{2}-c_{1}^{2}+c_{2}^{2}-c_{3}^{2}+dots dots+ )
( (-1)^{n} c_{n}^{2}=0, ) or ( (-1)^{frac{n}{2}} c_{frac{n}{2}} )
according as ( n ) is odd or even.
11
326The ( 3 r d ) term of ( (2+sqrt{3})^{3} ) is
A . 16
B. 17
c. 18
D. 19
11
327(a)
J
(0)
1
8.
Coefficient of 124 in (1 +12)12 (1+12) (1 + 24) is (20035)
(a) 12Cg +3 (b) 12Cq+1 (c) 12C ‘(d) 12C7+2
(2004)
21.
1
11
328Expand:
( left(frac{2}{x}-frac{x}{2}right)^{5} )
11
329( [mathrm{AS} 1] ) If ( boldsymbol{A}=frac{1}{3} boldsymbol{B} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{B}=frac{1}{2} boldsymbol{C}, ) then ( boldsymbol{A} )
( B: C= )
A .1: 3: 6
B. 2:3:6
( c cdot 3: 2: 6 )
D. 3: 1: 2
11
330Using binomial theorem evaluate the
following:
( (98)^{5} )
11
331Find the middle terms in the expansion
of ( left(2 x^{2}-frac{1}{x}right)^{7} )
В. ( -280 x^{5}, 560 x^{2} )
C ( .560 x^{5},-280 x^{2} )
D . ( 280 x^{5},-560 x^{2} )
11
332prove that ( C_{0}^{2 n} C_{n}-^{2 n-2} C_{n}+ )
( ^{2 n-4} C_{n}=2^{n} )where( C_{r}=^{n} C_{r} )
11
333The coefficient of ( 1 / x ) in the expansion of
( (1+x)^{n}(1+1 / x)^{n} ) is
A. ( frac{n !}{(n-1) !(n+1) !} )
B. ( frac{2 n !}{(n-1) !(n+1) !} )
c. ( frac{2 n !}{(2 n-1) !(2 n+1) !} )
D. none of these
11
334The middle term in the expansion of ( left(1-frac{1}{x}right)^{n}(1-x)^{n}, ) is
A ( cdot^{2 n} C_{n} )
в. ( ^{-2 n} C_{n} )
( mathrm{c} .^{-2 n} C_{n-1} )
D. none of these
11
335Using the formula for squaring a
binomial the value of ( (999)^{2} ) is:
A. 98009
B. 998005
c. 998001
D. 998002
11
336Number of rational terms in the
expansion of ( (sqrt{mathbf{2}}+sqrt{mathbf{3}})^{100} ) is
A . 25
B . 26
c. 27
D . 28
11
337If in the expansion of ( (1+x)^{n}, ) the
coefficients of three consecutive terms
are ( 56,70,56, ) then the value of ( n ) and
the position of the terms of these coefficients are given by
A ( cdot n=8, ) the terms are ( 4^{t h}, 5^{t h}, 6^{t h} )
B . ( n=7 ), the terms are ( 3^{r d}, 4^{t h}, 5^{t h} )
C ( cdot n=8 ), the terms are ( 5^{t h}, 6^{t h}, 7^{t h} )
D. ( n=7 ), the terms are ( 4^{t h}, 5^{t h}, 6^{t h} )
11
338The number of terms that are integers in the binomial expansion of ( (sqrt{7}+ ) ( sqrt{5})^{35} ) is
A . 4
B. 5
( c cdot 6 )
D.
11
339If sum of the first 3 coefficients is 16 in
the expansion ( left(x+frac{1}{x^{3}}right)^{n}, ) then find ( n )
A . 10
B. 8
( c .5 )
D.
11
340The coefficient of ( x^{5} ) in the expansion of
( (x+3)^{8} ) is
A . 1542
в. 1512
( c .2512 )
D. 12
E . 4
11
341WIL DULU
– 20.
T
The remainder left out when
out when 82n – (62)2n+1 is divided by 9
is:

(2) 2 (6) 7 (c) 8 (d) oh
11
342The coefficients of XP and x9 in the expansion of (1+x)pta
are

(a) equal
equal with opposite signs
reciprocals of each other
(d) none of these
11
343The term independent of ( x ) in the
binomial expansion of ( left(1-frac{1}{x}+3 x^{5}right)left(2 x^{2}-frac{1}{x}right)^{8} )
( mathbf{A} cdot-496 )
B . -400
c. 496
D. 400
11
344If the second term in the expansion ( left[a^{frac{1}{13}}+frac{a}{sqrt{a^{-1}}}right]^{n} ) is ( 14 a^{5 / 2} ), then the
value of ( frac{n C_{3}}{n C_{2}} ) is
( mathbf{A} cdot mathbf{4} )
B. 3
c. 12
D. 6
11
345Find the middle term of the expansion of ( left(sqrt{boldsymbol{x}}-frac{mathbf{1}}{boldsymbol{x}}right)^{mathbf{6}} )11
346Find the term independent of ( x ) in the
expansion of
( left(1+x+2 x^{2}right)left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} )
11
347In the expansion of ( (1+x)^{n} ) the
coefficients of ( p^{t h} ) and ( (p+1)^{t h} ) terms
are respectively ( p ) and ( q ) then ( p+q= )
( mathbf{A} cdot boldsymbol{n} )
B. ( n+1 )
( c cdot n+2 )
( mathbf{D} cdot n+3 )
11
348The coefficient of ( x ) in the expansion of
( left(1-x-x^{2}+x^{3}right)^{6} ) is ?
( A cdot 6 )
B. -6
c. -12
D. 12
11
349The fourth term in the expansion of ( left(p x+frac{1}{x}right)^{n} ) is ( frac{5}{2} . ) Then
This question has multiple correct options
( mathbf{A} cdot n=6 )
B . ( n=7 )
c. ( p=frac{1}{2} )
D. ( p=frac{1}{4} )
11
350If ( A ) is the coefficient of the middle term
in the expansion of ( (1+x)^{2 n} ) and ( B ) and
( mathrm{C} ) are the coefficients of two middle
terms in the expansion of ( (1+x)^{2 n-1} )
then
A. ( A+B=C )
в. ( B+C=A )
c. ( C+A=B )
D. ( A+B+C=0 )
11
351The total number of rational terms in
the expansion of ( (mathbf{7 3}+mathbf{1 1 9})^{mathbf{6 5 6 1}} )
A . 73
в. 729
( c .728 )
D. 730
E. 732
11
352The coeffecients of the middle term in
the binomial expansion in powers of ( x ) of ( (1+alpha x)^{4} ) and ( (1+alpha x)^{6} ) is the same
if ( boldsymbol{alpha} ) equals
A ( cdot-frac{5}{3} )
в. ( frac{10}{3} )
( c cdot frac{3}{10} )
D.
11
353The coefficient of ( x^{n} ) in the expansion of ( frac{1}{(1-x)(1-2 x)(1-3 x)} ) is
A ( cdot frac{1}{2}left(2^{n+2}-3^{n+3}+1right) )
B. ( frac{1}{2}left(3^{n+2}-2^{n+3}+1right) )
c. ( frac{1}{2}left(2^{n+3}-3^{n+2}+1right) )
D. none of these
11
354The middle term in the expansion of
( (1+x)^{2 n} ) is
A. ( frac{1.3 .5 ldots(2 n-1) 2^{n}}{n !} )
в. ( frac{1.2 .3 ldots(2 n-1) 2^{n} x^{n}}{n !} )
c. ( frac{1.3 .5 ldots(2 n-1) x^{n}}{n !} )
D. ( frac{1.3 .5 ldots .(2 n-1) 2^{n} x^{n}}{n !} )
11
355Find the term of the expansion of ( (a+ )
( b)^{50} ) which is the greatest in absolute
value if ( |boldsymbol{a}|=sqrt{mathbf{3}}|boldsymbol{b}| )
11
3561
2 Murks)
The sum of the rational terms in the expansion of
(2 + 31/5,10 is.
(1997 – 2 Marks)
11
357Fidn the ( 7^{t h} ) term from the end in the
expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 )
11
3581.
The larger of 9950 + 10050 and 10150 is
(1982-2 Marks)
7
11
359If ( (mathbf{1}+boldsymbol{x})^{mathbf{1 0}}=boldsymbol{a}_{mathbf{0}}+boldsymbol{a}_{mathbf{1}} boldsymbol{x}+boldsymbol{a}_{mathbf{2}} boldsymbol{x}^{mathbf{2}}+ )
( ldots ldots a_{10} x^{10}, ) then value of
( left(a_{0}-a_{2}+a_{4}-a_{6}+a_{8}-a_{10}right)^{2}+ )
( left(a_{1}-a_{3}+a_{5}-a_{7}+a_{9}right)^{2} ) is
( mathbf{A} cdot 2^{10} )
B . 2
( c cdot 2^{20} )
D. None of these
11
360Find the coefficient of: ( x ) in the
expansion of ( left(1-3 x+7 x^{2}right)(1-x)^{16} )
Enter 1 if answer is -19 otherwise enter
0
11
361If the coefficients of ( x^{2} ) and ( x^{3} ) are both
zero, in the expansion of the expression ( left(1+a x+b x^{2}right)(1-3 x)^{15} ) in powers of ( x )
then the ordered pair ( (a, b) ) is equal to:
A. (28,315)
B. (-54,315)
c. (-21,714)
D. (24,861)
11
362The number of integral terms in the expansion of ( (2 sqrt{5}+sqrt{7})^{642} )
A. 105
B. 107
( c .321 )
D. 108
11
363If the number of terms in the expansion
( (2 x+y)^{n}-(2 x-y)^{n} ) is ( 8, ) then the
value of ( n text { is } ldots ldots . . . . text { (where } n text { is odd }) )
A . 17
B. 19
c. 15
D. 13
11
364( (sqrt{2}+1)^{6}+(sqrt{2}-1)^{6}= )
A . 99
B. 98
c. 196
D. 198
11
365Number of terms in the expansion of ( left(x^{1 / 3}+x^{2 / 5}right)^{40} ) with integral power of ( x )
is equal to
11
366Compute the summation ( sum_{k=0}^{27} kleft(_{k}^{27}right)left(frac{1}{2}right)^{k}left(frac{2}{3}right)^{27-k} )11
367If the coefficient of the middle term in
the expansion of ( (1+x)^{2 n+2} ) is ( p ) and
the coefficients of middle terms in the
expansion of ( (1+x)^{2 n+1} ) are ( q ) and ( r )
then
A ( . p+q=r )
В. ( p+r=q )
( mathbf{c} cdot p=q+r )
D. ( p+q+r=0 )
11
368Expand the following binomial ( left(1+frac{x}{2}right)^{7} )11
369Find the cube of the following binomial expressions:
( 4-frac{1}{3 x} )
11
370If the middle term of ( (1+x)^{2 n} ) is the
greatest term then ( x ) lies between
A. ( n-1<x<n )
в. ( frac{n}{n+1}<x<frac{n+1}{n} )
c. ( n<x<n+1 )
D. ( frac{n+1}{n}<x<frac{n}{n+1} )
11

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